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Abstract and Applied Analysis
Volume 2014, Article ID 267384, 11 pages
Research Article

A Further Study of Almost Periodic Time Scales with Some Notes and Applications

1Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
2Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 2 June 2014; Revised 10 July 2014; Accepted 13 July 2014; Published 7 August 2014

Academic Editor: Elena Braverman

Copyright © 2014 Chao Wang and Ravi P. Agarwal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided. Furthermore, the concepts of almost periodic functions are redefined under the sense of this new timescale concept. Finally, almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. Besides, as an application, we apply our results to a class of high-order Hopfield neural networks.